Factoring Algebraic Expressions

The essence of factoring is to express a more complex expression as a product of simpler factors. This helps in solving equations and making algebraic expressions more transparent.

Extracting Common Factor

If every term in the expression has a common multiplicative factor, we can extract it.

The 3 is the common factor, extracting it gives a simpler form.

Notable Identities

Notable identities are frequently used rules that allow us to factor quickly.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a - b)(a + b)

Example: x² - 9 = (x - 3)(x + 3). This is the difference of squares formula.

Factoring Quadratic Expressions

Quadratic expressions are often written as products of two binomials. Example:

Here, the two numbers (2 and 3) are such that their product is 6 (the constant) and their sum is 5 (the coefficient of x).

Factoring in Everyday Life

Although it seems theoretical at first, factoring is useful in optimization tasks, area and volume calculations, or simplifying more complex equations.

• If a rectangle's area is x² + 5x + 6, factoring reveals (x + 2)(x + 3), so the sides could be x + 2 and x + 3 long.
• The a² - b² formula often appears in physics and engineering calculations when transforming squares of differences.

Practice Exercise